$\begingroup$

This is the definition of an increasing and decreasing function.

"A function $f(x)$ increases on an interval I if $f(b)≥f(a)\;\;\forall b>a$, where $a,b \in I$. If $f(b)>f(a) \;\;\forall \;\;b>a$, the function is said to be strictly increasing.

Conversely, A function $f(x)$ decreases on an interval I if $f(b)≤f(a)\;\;\forall b>a$, where $a,b \in I$. If $f(b)<f(a) \;\;\forall \;\;b>a$, the function is said to be strictly decreasing.

Then how would a horizontal line be described? If $f(x_2)=f(x_1)\;\; \forall x_2>x_1$ does that mean that a horizontal line meets the definition of an increasing function and a decreasing function?

$\endgroup$ 2

3 Answers

$\begingroup$

It means it is neither strictly increasing nor strictly decreasing, it is a constant function.

However, the way you have defined increasing and decreasing functions, the given horizontal line satisfies both definitions.

But it is better to avoid putting this type of function in any such class. It should be rightly referred to as a constant function.

$\endgroup$ 5 $\begingroup$

Yes, an horizontal line is a function $f$ such that $f(x)=c$ for all $x\in \mathbb{R}$, and, if $a\leq b\to f(a)=c\leq c =f(b)$. Idem for $\geq$

$\endgroup$ $\begingroup$

Inject $f(x)=c$ in the definitions:

"A function $c$ increases on an interval I if $c≥c\;\;\forall b>a$, where $a,b \in I$. If $c>c \;\;\forall \;\;b>a$, the function is said to be strictly increasing.

Conversely, A function $c$ decreases on an interval I if $c≤c\;\;\forall b>a$, where $a,b \in I$. If $c<c \;\;\forall \;\;b>a$, the function is said to be strictly decreasing."

You should be able to see for yourself what property holds.

$\endgroup$

Your Answer

Sign up or log in

Sign up using Google Sign up using Facebook Sign up using Email and Password Submit

Post as a guest

Post Your Answer Discard

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy